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arXiv · 1812.04930

Winding number and Cutting number of Harmonic cycle

Younng-Jin Kim·Woong Kook
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Abstract

A harmonic cycle $λ$, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces are isomorphic to the homology groups with real coefficients. In particular, if a cell complex has a one dimensional reduced homology, it has a unique harmonic cycle up to scalar, which we call the \emph{standard harmonic cycle}. In this paper, we will present a formula for the standard harmonic cycle $λ$ of a cell complex based on a high-dimensional generalization of cycletrees. Moreover, by using duality, we will define the standard harmonic cocycle $λ^*$, and show intriguing combinatorial properties of $λ$ and $λ^*$ in relation to (dual) spanning trees, (dual) cycletrees, winding numbers $w(\cdot)$ and cutting numbers $c(\cdot)$ in high dimensions.

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Younng-Jin Kim, Woong Kook. 2018-12-13. Winding number and Cutting number of Harmonic cycle. https://arxiv.org/abs/1812.04930

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